Method for measuring viscosity and/or elasticity of liquid

ABSTRACT

Viscosity and elasticity of a liquid are measured by immersing and vibrating a liquid tester in the liquid to be tested and measuring three frequency values that are a resonance frequency value on an amplitude characteristic curve obtained through vibration of the liquid tester in the liquid being tested, a low frequency value lower than the resonance frequency value on the amplitude characteristic curve at a phase angle smaller than a phase angle of 90 degrees at a resonance point on a phase angle characteristic curve obtained through the vibration in the liquid being tested, and a high frequency value higher than the resonance frequency value on the amplitude characteristic curve at a phase angle larger than the phase angle at the resonance point on the phase angle characteristic curve.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method for measuring the viscosityand/or elasticity of a liquid by which the viscoelasticity of the liquidto be tested can be accurately measured, and the viscosity or elasticitycan be accurately determined from the viscoelasticity.

2. Description of the Related Art

Japanese Patent Publication No. 4083621 discloses a method for measuringa viscosity value (viscosity). By this method, a vibrator (such as apiezoelectric device) is resonated in a liquid to be tested, and theviscosity value (the viscosity) of the liquid is measured from thedifference between the resonance frequency and either a low frequencyvalue (a half-value frequency f1) lower than the resonance frequencyvalue or a high frequency value (a half-value frequency f2) higher thanthe resonance frequency value.

Japanese Patent Publication No. 3348162 discloses a method for measuringthe viscoelasticity of a liquid. By this method, the frequency and theamplitude that are varied by the inherent viscoelasticity of the liquidare detected from the vibration of a liquid tester in the liquid to betested. The impedance of the liquid is determined from the frequency andthe amplitude, and the viscosity value and the elasticity value aredetermined from the real part and the imaginary part of the impedance.

By the viscosity measurement method disclosed in Japanese PatentPublication No. 4083621, however, a viscosity value is measured bydetermining the real part of the impedance of the subject liquid basedon the difference between the resonance frequency and either thehalf-value frequency f1 or the half-value frequency f2. By a measurementmethod involving only the real part of the impedance like this method,only the real part of complex viscosity can be determined. As a result,the viscosity value of a high viscosity liquid to be tested cannot bemeasured with accuracy, and it is impossible to measure both theviscosity value and the elasticity value that are required fordetermining the dynamic characteristics of the liquid.

By a measurement method involving determination of the impedance of aliquid from the frequency and the amplitude attributable to the inherentviscoelasticity of the liquid like the method disclosed in JapanesePatent Publication No. 3348162, or by a measurement method involving thereal part of the impedance as the reciprocal of a sensor output voltageproportional to the amplitude, the measurement accuracy becomes poorer,since the amplitude becomes smaller as the viscosity of the testedliquid becomes higher.

BRIEF SUMMARY OF THE INVENTION

The present invention provides a method for measuring the viscosityand/or elasticity of a liquid by which the viscoelasticity of the testedliquid can be accurately measured, and the viscosity or elasticity canbe precisely determined from the viscoelasticity.

A first measurement method according to the present invention is amethod for measuring the viscosity and/or elasticity of a liquid to betested by vibrating a liquid tester immersed in the liquid to be testedwith a drive source of a vibrator such as a piezoelectric device, or byimmersing and vibrating the vibrator as the liquid tester in the liquidto be tested. This method includes: measuring three frequency valuesthat are the resonance frequency value (f0) on the amplitudecharacteristic curve obtained through the vibration of the liquid testerin the liquid being tested, a low frequency value (f1) lower than theresonance frequency value (f0) on the amplitude characteristic curve,and a high frequency value (f2) higher than the resonance frequencyvalue (f0) on the amplitude characteristic curve; calculating the realpart of the impedance of the liquid being tested, using the highfrequency value (f2) and the low frequency value (f1); calculating theimaginary part of the impedance of the liquid being tested, using theresonance frequency value (f0); and calculating the viscosity valueand/or the elasticity value of the liquid being tested from the realpart and the imaginary part of the impedance.

A second measurement method according to the present invention alsoincludes: measuring three frequency values that are the resonancefrequency value (f0) on the amplitude characteristic curve obtainedthrough the vibration of the liquid tester in the liquid being tested, alow frequency value (f1) lower than the resonance frequency value (f0)on the amplitude characteristic curve, and a high frequency value (f2)higher than the resonance frequency value (f0) on the amplitudecharacteristic curve; calculating the real part and the imaginary partof the impedance of the liquid being tested, using the three frequencyvalues that are the resonance frequency value (f0), the low frequencyvalue (f1), and the high frequency value (f2); and calculating theviscosity value and/or the elasticity value of the liquid being testedfrom the real part and the imaginary part of the impedance.

Preferably, the low frequency value (f1) and the high frequency value(f2) represent the frequencies at symmetrical phase angles with respectto the resonance point on the phase angle characteristic curve obtainedthrough the vibration in the liquid being tested.

By the measurement method according to the present invention, the realpart and the imaginary part of impedance are determined with the use ofthe resonance frequency (f0), the low frequency value (f1), and the highfrequency value (f2). Even if the liquid to be tested is a highviscosity liquid, the viscoelasticity of the liquid can be accuratelymeasured, and the viscosity or elasticity can be accurately determinedfrom the viscoelasticity.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the phase angle characteristic curves observed when aliquid tester is vibrated in a liquid to be tested;

FIG. 2 shows the amplitude characteristic curves observed when theliquid tester is vibrated in the liquid to be tested;

FIGS. 3A and 3B are perspective views showing the vibration modes forthe liquid tester in vibrator devices that include a drive source in theliquid tester immersed in the liquid to be tested; and

FIG. 4 is a perspective view showing a vibrator device that vibrates theliquid tester with a drive source provided outside the tested liquid,and the vibration mode for the liquid tester.

DETAILED DESCRIPTION OF THE INVENTION

The following is a description of preferred embodiments of the presentinvention, with reference to FIGS. 1 through 4.

In FIG. 1, G1 is the characteristic curve of the delay angle of theamplitude with respect to the vibrational frequency of a liquid tester 1that vibrates in a liquid to be tested, or the characteristic curve ofthe phase angle.

On the other hand, G2 is the characteristic curve of the delay angle ofthe amplitude with respect to the vibrational frequency observed whenthe liquid tester 1 is vibrated in the air, or the characteristic curveof the phase angle.

In FIG. 2, G3 is the characteristic curve of the amplitude with respectto the vibrational frequency of the liquid tester 1 that vibrates in theliquid to be tested.

On the other hand, G4 is the characteristic curve of the amplitude withrespect to the vibrational frequency observed when the liquid tester 1is vibrated in the air.

In FIG. 1, f0, f1, and f2 on the phase angle characteristic curves G1and G2 represent the frequencies at the respective phase angles of 90degrees, 90−d degrees, and 90+d degrees. In this case, f0 represents theresonance frequency, f1 represents a frequency lower than the resonancefrequency, and f2 represents a frequency higher than the resonancefrequency.

Here, −d is the angle to be subtracted from 90 degrees, and +d is theangle to be added to 90 degrees. For example, d represents an angle of45 degrees of symmetric point with respect to 90 degrees, and in thiscase, 90−d degrees is 45 degrees, and 90+d degrees is 135 degrees.

In this embodiment, f1 and f2 are used as example frequencies atsymmetrical phase angles with respect to the resonance point existing onthe phase angle characteristic curves obtained through vibrations of theliquid to be tested.

Meanwhile, f0, f1, and f2 on the amplitude characteristic curves G3 andG4 in FIG. 2 represent the frequencies on the amplitude characteristiccurves equivalent to f0, f1, and f2 on the phase angle characteristiccurves G1 and G2. Here, f0 represents the resonance frequency, f1represents a frequency lower than the resonance frequency, and f2represents a frequency higher than the resonance frequency.

A vibrator device including the liquid tester 1 in accordance with thepresent invention is either a device that vibrates (or resonates) theliquid tester 1 by transmitting the vibration of a drive source 2 formedwith a piezoelectric device or an electromagnetic drive device to bedriven outside the tested liquid via a vibration transmitting shaft 3 asshown in FIG. 4, or a device that vibrates the liquid tester 1 that is avibrator formed with a piezoelectric device immersed in the liquid to betested as disclosed in Japanese Patent Publication No. 4083621.

The vibrator device may also be a device that includes the drive source2 formed with a piezoelectric device in the liquid tester 1, andvibrates the liquid tester 1 in the liquid to be tested, as shown inFIGS. 3A and 3B.

Examples vibration modes for the liquid tester 1 include the mode forvibrating the liquid tester 1 in a circle direction about the shaft lineas shown in FIG. 3A, and the mode for vibrating the liquid tester 1 in atransverse direction in the liquid to be tested as shown in FIG. 3B.

The present invention relates to a method for measuring the viscosityvalue and/or elasticity value of a liquid to be tested by immersing theliquid tester 1 that is vibrated by a vibrator in the liquid to betested, as described above. By a first measurement method according tothe present invention, the three frequency values, which are theresonance frequency value f0 on the phase angle characteristic curve (G1in FIG. 1) and the amplitude characteristic curve (G3 in FIG. 2)obtained through the vibration of the liquid tester 1 in the liquid tobe tested, the low frequency value f1 lower than the resonance frequencyvalue f0 on the amplitude characteristic curve, and the high frequencyvalue f2 higher than the resonance frequency value f0 on the amplitudecharacteristic curve, are first measured. The real part of the impedanceof the tested liquid is calculated with the use of the high frequencyvalue f2 and the low frequency value f1, and the imaginary part of theimpedance of the tested liquid is calculated with the use of theresonance frequency value f0. The viscosity value and/or elasticityvalue of the tested liquid are then calculated from the real part andimaginary part of the impedance.

By a second measurement method according to the present invention, thethree frequency values, which are the resonance frequency value f0 onthe amplitude characteristic curve obtained through the vibration of theliquid tester 1 in the liquid to be tested, the low frequency value f1lower than the resonance frequency value f0 on the amplitudecharacteristic curve, and the high frequency value f2 higher than theresonance frequency value f0 on the amplitude characteristic curve, arefirst measured. The real part and imaginary part of the impedance of thetested liquid are calculated with the use of the three frequency values:the resonance frequency value f0, the low frequency value f1, and thehigh frequency value f2. The viscosity value and/or elasticity value ofthe tested liquid are then calculated from the real part and imaginarypart of the impedance.

It is preferable that the low frequency value f1 and the high frequencyvalue f2 are the frequencies at phase angles located at the symmetricalpoints with respect to the resonance point on the phase anglecharacteristic curve G1 obtained through the vibration in the testedliquid.

In other words, the low frequency value f1 and the high frequency valuef2 are the frequencies at the symmetrical points located at the sameamplitude level with respect to the resonance point on the amplitudecharacteristic curve G3 obtained through the vibration in the testedliquid.

As can be seen from the comparison between the phase anglecharacteristic curves G1 and G2 in FIG. 1, and the comparison betweenthe amplitude characteristic curves G3 and G4 in FIG. 2, the resonancefrequency value f0 in the tested liquid is lower than the resonancefrequency value f0 in the air, and the difference between the highfrequency value f2 and the low frequency value f1 in the tested liquidis larger than the difference between the high frequency value f2 andthe low frequency value f1 in the air.

In other words, as the viscosity of the tested liquid becomes greater,the resonance frequency value f0 shift to the lower frequency side, andthe difference between the high frequency value f2 and the low frequencyvalue f1 becomes greater.

The following is a more detailed description of the first and secondmethods for measuring the viscosity and/or elasticity of a liquid inaccordance with the present invention, with reference to equations (1)through (47).

The delays in the vibration amplitude with respect to a drive signal arerepresented by phase angles d0, d1, and d2. As f0 represents theresonance point, d0 is 90 degrees. At f1 and f2, d1 is 90−d degrees, d2is 90+d degrees, and the angular frequencies of f0, f1, and f2 arerepresented by ω₀, ω₁, and ω₂. Here, ω is equal to 2πf.

The angular frequencies at f0, f1, and f2 in the air are represented byω₀₀, ω₀₁, and ω₀₂, respectively. Here, ω₀₀, ω₀₁, and ω₀₂ are inherentvalues of the vibrator devices shown in FIGS. 3A, 3B, and 4, and areconstants that are measured in advance and are stored in a memory of aprocessor provided in each of the vibrator devices. The processor hascalculating functions.

The liquid tester 1 vibrates a liquid having viscosity η and density ρat the resonance angular frequency ω₀ of the liquid tester 1. The ratioF/ν between the vibration velocity ν of the tested liquid and theresistive force F generated from the tested liquid is called theimpedance Z of the tested liquid.

The vibration velocity ν of the waves propagated to the tested liquidcan be determined as the solution of a wave equation. The impedance Z ofthe tested liquid is determined according to the following equation (1)based on the vibration velocity ν determined by plane waveapproximation. Here, S represents a constant proportional to the surfacearea of the liquid tester 1, and j represents the symbol of an imaginarypart.Z=S·√{square root over (jω·ρ·η)}  Equation (1):

Where the equation (1) is Maclaurin-expanded in the vicinity of ω₀ so asto express the impedance Z with ω₀, the impedance Z of the tested liquidcan be expressed by the following equation (2) when ω₂ is very close toω₀.Z=S·√{square root over (jω ₀·ρ·η)}  Equation (2):

When ω₁ and ω₂ greatly differ from ω₀, the impedance Z of the testedliquid is expressed by the following equation (3). Here, ω is ω₀, ω₁, orω₂.

$\begin{matrix}{Z = {S \cdot \sqrt{{j\omega}_{0} \cdot \rho \cdot \eta} \cdot \frac{\omega + \omega_{0}}{2 \cdot \omega_{0}}}} & {{Equation}\mspace{14mu}(3)}\end{matrix}$

If the tested liquid is a viscoelastic material, the viscosity η of thetested liquid is a complex number, and can be expressed by the followingequation (4).

$\begin{matrix}\begin{matrix}{\eta = {n^{\prime} - {j \cdot \eta^{''}}}} \\{= {\sqrt{\eta^{\prime 2} + \eta^{''2}} \cdot {\mathbb{e}}^{- {j\delta}}}}\end{matrix} & {{Equation}\mspace{14mu}(4)}\end{matrix}$

The phase angle δ of the complex viscosity is defined like the equation(5).

$\begin{matrix}{{\tan\;\delta} = \frac{\eta^{''}}{\eta^{\prime}}} & {{Equation}\mspace{14mu}(5)}\end{matrix}$

The absolute value η₁ of the complex viscosity is defined like theequation (6).√{square root over (η′²+η″²)}η₁  Equation (6):

The equation (4) can be expressed by the equation (7) using η₁.η=η₁ ·e ^(−jδ)  Equation (7):

The equations (1) through (7) are common between the first and secondmethods for measuring the viscosity and/or elasticity of a liquid. Inthe following, the calculations of the real part R and the imaginarypart X of the impedance Z according to the first measurement method aredescribed with the use of equations (8) through (29).

When ω₁ and ω₂ are very close to ω₀, or when the impedance Z of thetested liquid is almost the same at the three measurement values, theimpedance Z is determined by assigning the equation (7) to the equation(2), and is expressed by the following equations (8) and (9).

$\begin{matrix}\begin{matrix}{Z = {S \cdot \sqrt{{j\omega}_{0} \cdot \rho \cdot \eta_{l} \cdot {\mathbb{e}}^{- {j\delta}}}}} \\{= {{S \cdot \sqrt{\omega_{0} \cdot \rho \cdot \eta_{l}} \cdot {\cos\left( {\frac{\pi}{4} - \frac{\delta}{2}} \right)}} +}} \\{{~~}{j \cdot S \cdot \sqrt{\omega_{0} \cdot \rho \cdot \eta_{l}} \cdot {\sin\left( {\frac{\pi}{4} - \frac{\delta}{2}} \right)}}}\end{matrix} & \begin{matrix}{{Equation}\mspace{14mu}(8)} \\{{Equation}\mspace{14mu}(9)}\end{matrix}\end{matrix}$

Where the real part R and the imaginary part X of the impedance Z aredefined by the equations (10) and (11), the impedance Z can be expressedby the equation (12).

$\begin{matrix}{R = {S \cdot \sqrt{\omega_{0} \cdot \rho \cdot \eta_{l}} \cdot {\cos\left( {\frac{\pi}{4} - \frac{\delta}{2}} \right)}}} & {{Equation}\mspace{14mu}(10)} \\{X = {S \cdot \sqrt{\omega_{0} \cdot \rho \cdot \eta_{l}} \cdot {\sin\left( {\frac{\pi}{4} - \frac{\delta}{2}} \right)}}} & {{Equation}\mspace{14mu}(11)} \\{Z = {R + {j \cdot X}}} & {{Equation}\mspace{14mu}(12)}\end{matrix}$

The inertia moment of the liquid tester 1 is represented by M, thetorsion spring constant is represented by K, and the internal resistanceof the torsion spring constant is represented by r.

When the liquid tester 1 is driven by a drive force F=F₀·e^(jωt) in theair, the displacement of the liquid tester 1 is χ=χ₀·e^(jωt). Thedisplacement can be expressed by the vibration equation according to thefollowing equation (13).−ω² ·M·x ₀ +j·ω·r·x ₀ +K·x ₀ =F ₀  Equation (13):

When the liquid tester 1 is immersed in the tested liquid, the real partR and the imaginary part X of the impedance Z of the tested liquidrespectively act as an addition to the resistance and an addition to theinertia moment in the vibrator device.

Accordingly, the internal resistance r and the inertia moment M becomer+R and M+X/ω, respectively, and the equation (13) turns into thefollowing equation (14).

$\begin{matrix}{{{{- \omega^{2}} \cdot \left( {M + \frac{X}{\omega}} \right) \cdot x_{0}} + {j \cdot \omega \cdot \left( {r + R} \right) \cdot x_{0}} + {K \cdot x_{0}}} = F_{0}} & {{Equation}\mspace{14mu}(14)}\end{matrix}$

By transforming the equation (14), the transmission function χ₀/F₀ withrespect to the drive force F of the movement of the liquid tester 1 canbe expressed by the equation (15).

$\begin{matrix}{\frac{x_{0}}{F_{0}} = \frac{1}{{{- \omega^{2}} \cdot \left( {M + \frac{X}{\omega}} \right)} + K + {j \cdot \omega \cdot \left( {r + R} \right)}}} & {{Equation}\mspace{14mu}(15)}\end{matrix}$

With ω₀₀ ² being equal to K/M, the denominator of the equation (15) isrationalized to find the equation (16).

$\begin{matrix}\begin{matrix}{\frac{x_{0}}{F_{0}} = {\frac{1}{M} \cdot \frac{1}{\omega_{00}^{2} - {\omega^{2} \cdot \left( {1 + \frac{X}{\omega \cdot M}} \right)} + {j \cdot \omega \cdot \frac{r + R}{M}}}}} \\{{= {\frac{1}{M} \cdot \frac{\omega_{00}^{2} - {\omega^{2} \cdot \left( {1 + \frac{X}{\omega \cdot M}} \right)} - {j \cdot \omega \cdot \frac{r + R}{M}}}{\left\{ {\omega_{00}^{2} - {\omega^{2} \cdot \left( {1 + \frac{X}{\omega \cdot M}} \right)}} \right\}^{2} + \left( {\omega \cdot \frac{r + R}{M}} \right)^{2}}}},}\end{matrix} & {{Equation}\mspace{14mu}(16)}\end{matrix}$

Where D represents the phase angle indicating the delay of the movementof the liquid tester 1 with respect to the drive force, tan D isexpressed by the equation (17).

$\begin{matrix}{{\tan\; D} = \frac{\omega \cdot \frac{r + R}{M}}{\omega_{00}^{2} - {\omega^{2} \cdot \left( {1 + \frac{X}{\omega \cdot M}} \right)}}} & {{Equation}\mspace{14mu}(17)}\end{matrix}$

Here, D represents the phase angles at the three points d₁, d₂, and d₀on the phase angle characteristic curves shown in FIG. 1. Since theangular frequencies at these three points are ω₁, ω₂, and ω₀, the threeequations formed with the three sets (d₁, ω₁), (d₂, ω₂), and (d₀, ω₀)are the following equations (18), (19), and (20).

$\begin{matrix}{{\tan\; d_{1}} = \frac{\omega_{1} \cdot \frac{r + R}{M}}{\omega_{00}^{2} - {\omega_{1}^{2} \cdot \left( {1 + \frac{X}{\omega_{1} \cdot M}} \right)}}} & {{Equation}\mspace{14mu}(18)} \\{{\tan\; d_{2}} = \frac{\omega_{2} \cdot \frac{r + R}{M}}{\omega_{00}^{2} - {\omega_{2}^{2} \cdot \left( {1 + \frac{X}{\omega_{2} \cdot M}} \right)}}} & {{Equation}\mspace{14mu}(19)} \\{{\omega_{00}^{2} - {\omega_{0}^{2} \cdot \left( {1 + \frac{X}{\omega_{0} \cdot M}} \right)}} = 0} & {{Equation}\mspace{14mu}(20)}\end{matrix}$

Since d₁ is 90−d, and d₂ is 90+d, tan d₁ is equal to cot d, and tan d₂is equal to −cot d. Here, cot d is 1/C. With the use of C, the equations(18), (19), and (20) are transformed into the following equations (21),(22), and (23).

$\begin{matrix}{{\omega_{00}^{2} - \omega_{1}^{2} - {\frac{X}{M} \cdot \omega_{1}}} = {C \cdot \omega_{1} \cdot \frac{r + R}{M}}} & {{Equation}\mspace{14mu}(21)} \\{{\omega_{00}^{2} - \omega_{2}^{2} - {\frac{X}{M} \cdot \omega_{2}}} = {{- C} \cdot \omega_{2} \cdot \frac{r + R}{M}}} & {{Equation}\mspace{14mu}(22)} \\{{\omega_{00}^{2} - \omega_{0}^{2} - {\frac{X}{M} \cdot \omega_{0}}} = 0} & {{Equation}\mspace{14mu}(23)}\end{matrix}$

The following equation (24) is obtained from the equations (21) and(22).

$\begin{matrix}{{{\omega_{00}^{2} \cdot \left( {\omega_{2} - \omega_{1}} \right)} - \left( {{\omega_{1}^{2} \cdot \omega_{2}} - {\omega_{1} \cdot \omega_{2}^{2}}} \right)} = {2 \cdot C \cdot \omega_{1} \cdot \omega_{2} \cdot \frac{r + R}{M}}} & {{Equation}\mspace{14mu}(24)}\end{matrix}$

The following equation (25) expressing r+R/M is determined from theequation (24).

$\begin{matrix}{\frac{r + R}{M} = \frac{\left( {\omega_{2} - \omega_{1}} \right) \cdot \left( {\omega_{00}^{2} + {\omega_{1} \cdot \omega_{2}}} \right)}{2 \cdot C \cdot \omega_{1} \cdot \omega_{2}}} & {{Equation}\mspace{14mu}(25)}\end{matrix}$

Also, the equation (26) expressing X/M is determined from the equation(23).

$\begin{matrix}{\frac{X}{M} = \frac{\omega_{00}^{2} - \omega_{0}^{2}}{\omega_{0}}} & {{Equation}\mspace{14mu}(26)}\end{matrix}$

In the air, ω₁, ω₂, and ω₀ are the known constants ω₀₁, ω₀₂, and ω₀₀,and are stored in the memory of the processor for processing signalsobtained form the vibrator device.

Since R is 0, and X is 0 in the air, the equation (25) is transformedinto the following equation (27).

$\begin{matrix}{\frac{r}{M} = \frac{\left( {\omega_{02} - \omega_{01}} \right) \cdot \left( {\omega_{00}^{2} + {\omega_{01} \cdot \omega_{02}}} \right)}{2 \cdot C \cdot \omega_{01} \cdot \omega_{02}}} & {{Equation}\mspace{14mu}(27)}\end{matrix}$

As is apparent from the equation (27), r/M is a known constant. With theuse of r/M of the equation (27), the real part R and the imaginary partX can be determined according to the following equations (28) and (29).

$\begin{matrix}{R = {{\frac{M}{2 \cdot C} \cdot \frac{\left( {\omega_{2} - \omega_{1}} \right) \cdot \left( {\omega_{00}^{2} + {\omega_{1} \cdot \omega_{2}}} \right)}{\omega_{1} \cdot \omega_{2}}} - r}} & {{Equation}\mspace{14mu}(28)} \\{X = {M \cdot \frac{\left( {\omega_{00} - \omega_{0}} \right) \cdot \left( {\omega_{00} + \omega_{0}} \right)}{\omega_{0}}}} & {{Equation}\mspace{14mu}(29)}\end{matrix}$

Next, the calculations of the real part R and the imaginary part X ofthe impedance Z according to the second measurement method are describedwith the use of equations (30) through (33).

When ω₁ and ω₂ greatly differ from ω₀, the impedance Z of the testedliquid is expressed by the equation (3). Since the internal resistance rof the device is considered to be small enough to ignore, the equations(21) and (22) are transformed into the following equations (30) and(31).

$\begin{matrix}{{\omega_{00}^{2} - \omega_{1}^{2} - {\frac{X}{M} \cdot \omega_{1} \cdot \frac{\omega_{1} + \omega_{0}}{2 \cdot \omega_{0}}}} = {C \cdot \omega_{1} \cdot \frac{\omega_{1} + \omega_{0}}{2 \cdot \omega_{0}} \cdot \frac{R}{M}}} & {{Equation}\mspace{14mu}(30)} \\{{\omega_{00}^{2} - \omega_{2}^{2} - {\frac{X}{M} \cdot \omega_{2} \cdot \frac{\omega_{2} + \omega_{0}}{2 \cdot \omega_{0}}}} = {{- C} \cdot \omega_{2} \cdot \frac{\omega_{2} + \omega_{0}}{2 \cdot \omega_{0}} \cdot \frac{R}{M}}} & {{Equation}\mspace{14mu}(31)}\end{matrix}$

According to the equations (30) and (31), the real part R and theimaginary part X are expressed by the following equations (32) and (33).

$\begin{matrix}{R = {M \cdot \frac{\begin{matrix}{\omega_{00} \cdot \left( {\omega_{2} - \omega_{1}} \right) \cdot} \\\left\{ {\omega_{00}^{2} + {\omega_{1} \cdot \omega_{2}} + {\omega_{0} \cdot \left( {\omega_{1} + \omega_{2}} \right)}} \right\}\end{matrix}}{C \cdot \omega_{1} \cdot \omega_{2} \cdot \left( {\omega_{1} + \omega_{2}} \right) \cdot \left( {\omega_{2} + \omega_{0}} \right)}}} & {{Equation}\mspace{14mu}(32)} \\{X = {M \cdot \frac{\begin{matrix}{{\omega_{00}^{2} \cdot \left( {\omega_{1} + \omega_{2}} \right)} + {2 \cdot \omega_{00}^{2} \cdot \omega_{0}} -} \\{{\omega_{1} \cdot \omega_{2} \cdot \left( {\omega_{1} + \omega_{2}} \right)} - {\omega_{0}^{2} \cdot \left( {\omega_{1}^{2} + \omega_{2}^{2}} \right)}}\end{matrix}}{C \cdot \omega_{1} \cdot \omega_{2} \cdot \left( {\omega_{1} + \omega_{2}} \right) \cdot \left( {\omega_{2} + \omega_{0}} \right)}}} & {{Equation}\mspace{14mu}(33)}\end{matrix}$

As described above, by the first measurement method, the liquid tester 1is vibrated in the tested liquid, and the three angular frequencies ω₁,ω₂, and ω₀ at the three phase differences d₁, d₂, and d₀ are measured.The real part R of the impedance Z of the tested liquid is calculated bythe processor according to the equation (28) using the high frequencyvalue f2 and the low frequency value f1. The imaginary part X of theimpedance Z of the tested liquid is calculated according to the equation(29) using the resonance frequency value f0.

By the second measurement method, the real part R and the imaginary partX of the impedance Z of the tested liquid are calculated according tothe equations (32) and (33) using the high frequency value f2, the lowfrequency value f1, and the resonance frequency value f0.

Next, the first measurement method for measuring the viscosity and/orelasticity of the liquid based on the real part R and the imaginary partX determined according to the equations (28) and (29) is described withthe use of equations (34) through (40).

Based on the equations (10) and (11), the phase angle δ of the complexviscosity is calculated according to the following equation (34).

$\begin{matrix}{\delta = {\frac{\pi}{2} - {2 \cdot {\arctan\left( \frac{X}{R} \right)}}}} & {{Equation}\mspace{14mu}(34)}\end{matrix}$

Since R²+X²=S²·ω₀·ρ·η₁ according to the equations (10) and (11), theabsolute value η₁ of the complex viscosity is calculated according tothe following equation (35) using R and X determined by the equations(28) and (29).

$\begin{matrix}{\eta_{l} = \frac{R^{2} + X^{2}}{S^{2} \cdot \omega_{0} \cdot \rho}} & {{Equation}\mspace{14mu}(35)}\end{matrix}$

Based on the phase angle δ of the complex viscosity determined by theequation (34) and the absolute value η₁ of the complex viscositydetermined by the equation (35), the complex viscosity η is calculatedaccording to the following equation (36).

$\begin{matrix}\begin{matrix}{\eta = {\eta_{l} \cdot {\mathbb{e}}^{- {j\delta}}}} \\{= {{{\eta_{l} \cdot \cos}\;\delta} - {{j \cdot \eta_{l} \cdot \sin}\;\delta}}} \\{= {\eta^{\prime} - {j \cdot \eta^{''}}}} \\{= {n^{\prime} - {j \cdot \frac{G^{\prime}}{\omega_{0}}}}}\end{matrix} & {{Equation}\mspace{14mu}(36)}\end{matrix}$

Based on the equation (36), the real part η′ of the complex viscosity ηis calculated according to the following equation (37).η′=η₁·cos δ  Equation (37):

The imaginary part η″ of the complex viscosity η is calculated accordingto the following equation (38).η″=η₁·sin δ  Equation (38):

The real part G′ of the complex elastic modulus G is calculatedaccording to the following equation (39).G′=ω ₀·η₁·sin δ  Equation (39):

The imaginary part G″ of the complex elastic modulus G is calculatedaccording to the following equation (40).G″=ω ₀·η₁·cos δ  Equation (40):

Next, the second measurement method for measuring the viscosity and/orelasticity of the liquid based on the real part R and the imaginary partX determined according to the equations (32) and (33) is described withthe use of equations (41) through (47).

Even where the impedance Z is expressed by the equation (3), the realpart R and the imaginary part X of the impedance Z at the resonancefrequency value are expressed by the equations (10) and (11). Based onthe equations (10) and (11), the phase angle δ of the complex viscosityis calculated according to the following equation (41).

$\begin{matrix}{\delta = {\frac{\pi}{2} - {2 \cdot {{\arctan\left( \frac{X}{R} \right)}.}}}} & {{Equation}\mspace{14mu}(41)}\end{matrix}$

Since R²+X²=S²·ω₀·ρ·η₁ according to the equations (10) and (11), theabsolute value η₁ of the complex viscosity is calculated according tothe following equation (42) using R and X determined by the equations(32) and (33).

$\begin{matrix}{\eta_{l} = \frac{R^{2} + X^{2}}{S^{2} \cdot \omega_{0} \cdot \rho}} & {{Equation}\mspace{14mu}(42)}\end{matrix}$

Based on the phase angle δ of the complex viscosity determined by theequation (41) and the absolute value η₁ of the complex viscositydetermined by the equation (42), the complex viscosity η is calculatedaccording to the following equation (43).

$\begin{matrix}\begin{matrix}{\eta = {\eta_{l} \cdot {\mathbb{e}}^{- {j\delta}}}} \\{= {{{\eta_{l} \cdot \cos}\;\delta} - {{j \cdot \eta_{l} \cdot \sin}\;\delta}}} \\{= {\eta^{\prime} - {j \cdot \eta^{''}}}} \\{= {n^{\prime} - {j \cdot \frac{G^{\prime}}{\omega_{0}}}}}\end{matrix} & {{Equation}\mspace{14mu}(43)}\end{matrix}$

Based on the equation (43), the real part η′ of the complex viscosity ηis calculated according to the following equation (44).η′=η₁·cos δ  Equation (44):

The imaginary part η″ of the complex viscosity η is calculated accordingto the following equation (45).η″=η₁·sin δ  Equation (45):

The real part G′ of the complex elastic modulus G is calculatedaccording to the following equation (46).G′=ω ₀·η₁·sin δ  Equation (46):

The imaginary part G″ of the complex elastic modulus G is calculatedaccording to the following equation (47).G″=ω ₀·η₁·cos δ  Equation (47):

As described above, by the first measurement method, the three frequencyvalues, which are the resonance frequency value f0 on the phase anglecharacteristic curve (G1 in FIG. 1) and the amplitude characteristiccurve (G3 in FIG. 2) obtained through vibration of the liquid tester 1in a liquid to be tested, the low frequency value f1 lower than theresonance frequency value f0 on the amplitude characteristic curve, andthe high frequency value f2 higher than the resonance frequency value f0on the amplitude characteristic curve, are first measured. The real partof the impedance of the tested liquid is calculated with the use of thehigh frequency value f2 and the low frequency value f1, and theimaginary part of the impedance of the tested liquid is calculated withthe use of the resonance frequency value f0. The viscosity value and/orthe elasticity value of the tested liquid are calculated from the realpart and imaginary part of the impedance. By the second measurementmethod, the three frequency values, which are the resonance frequencyvalue f0 on the amplitude characteristic curve obtained throughvibration of the liquid tester 1 in a liquid to be tested, the lowfrequency value f1 lower than the resonance frequency value f0 on theamplitude characteristic curve, and the high frequency value f2 higherthan the resonance frequency value f0 on the amplitude characteristiccurve, are first measured. The real part and the imaginary part of theimpedance of the tested liquid are calculated with the use of the threefrequency values, which are the resonance frequency value f0, the lowfrequency value f1, and the high frequency value f2. The viscosity valueand/or the elasticity value of the tested liquid are calculated from thereal part and imaginary part of the impedance. By the first and secondmeasurement methods, accurate viscoelasticity measurement can beperformed on a wide variety of liquids ranging from low viscosityliquids to high viscosity liquids to be tested. Furthermore, viscosityor elasticity can be accurately determined from the viscoelasticity.

Accordingly, it is possible to effectively solve the problems of theconventional methods, such as the problem that the real part and theimaginary part of complex viscosity cannot be determined, and thereforethe viscosity value of a high viscoelasticity liquid to be tested cannotbe accurately measured as in a case where only the real part of theimpedance is used as disclosed in Japanese Patent Publication No.4083621, and the problem that it is impossible to measure both theviscosity value and the elasticity value required for determining thedynamic characteristics of the liquid.

Also, it is possible to effectively solve the problems of theconventional methods, such as the problem that the amplitude becomessmaller as the viscosity of the tested liquid becomes higher, andtherefore the measurement accuracy becomes poorer, as in a case wherethe impedance of the liquid is determined from the frequency andamplitude attributable to the inherent viscoelasticity of the liquid, orwhere the real part of the impedance is regarded as the reciprocal ofthe sensor output voltage proportional to the amplitude as disclosed inJapanese Patent Publication No. 3348162.

1. A method for measuring viscosity and elasticity of a liquid byimmersing and vibrating a liquid tester in the liquid to be tested, themethod comprising: measuring three frequency values that are a resonancefrequency value (f0) on an amplitude characteristic curve obtainedthrough vibration of the liquid tester in the liquid being tested, a lowfrequency value (f1) lower than the resonance frequency value (f0) onthe amplitude characteristic curve at a phase angle smaller than a phaseangle of 90 degrees at a resonance point on a phase angle characteristiccurve obtained through the vibration in the liquid being tested, and ahigh frequency value (f2) higher than the resonance frequency value (f0)on the amplitude characteristic curve at a phase angle larger than thephase angle at the resonance point on the phase angle characteristiccurve; calculating a real part of impedance of the liquid being testedusing the high frequency value (f2) and the low frequency value (f1);calculating an imaginary part of the impedance of the liquid beingtested using the resonance frequency value (f0); and calculating theviscosity value and the elasticity value of the liquid being tested fromthe real part and the imaginary part of the impedance.
 2. A method formeasuring viscosity and elasticity of a liquid by immersing andvibrating a liquid tester in the liquid to be tested, the methodcomprising: measuring three frequency values that are a resonancefrequency value (f0) on an amplitude characteristic curve obtainedthrough vibration of the liquid tester in the liquid being tested, a lowfrequency value (f1) lower than the resonance frequency value (f0) onthe amplitude characteristic curve at a phase angle smaller than a phaseangle of 90 degrees at a resonance point on a phase angle characteristiccurve obtained through the vibration in the liquid being tested, and ahigh frequency value (f2) higher than the resonance frequency value (f0)on the amplitude characteristic curve at a phase angle larger than thephase angle at the resonance point on the phase angle characteristiccurve; calculating a real part and an imaginary part of impedance of theliquid being tested using the three frequency values that are theresonance frequency value (f0), the low frequency value (f1), and thehigh frequency value (f2); and calculating the viscosity value and theelasticity value of the liquid being tested from the real part and theimaginary part of the impedance.
 3. The method of claim 1, wherein thelow frequency value (f1) and the high frequency value (f2) representfrequencies at symmetrical phase angles with respect to a resonancepoint on the phase angle characteristic curve obtained through thevibration in the liquid being tested.
 4. The method of claim 2, whereinthe low frequency value (f1) and the high frequency value (f2) representfrequencies at symmetrical phase angles with respect to a resonancepoint on the phase angle characteristic curve obtained through thevibration in the liquid being tested.
 5. The method of claim 1, furthercomprising: immersing the liquid tester in the liquid; and vibrating theliquid tester in the liquid.
 6. The method of claim 5, wherein the lowfrequency value (f1) and the high frequency value (f2) representfrequencies at symmetrical phase angles with respect to a resonancepoint on the phase angle characteristic curve obtained through thevibration in the liquid being tested.
 7. The method of claim 2, furthercomprising: immersing the liquid tester in the liquid; and vibrating theliquid tester in the liquid.
 8. The method of claim 6, wherein the lowfrequency value (f1) and the high frequency value (f2) representfrequencies at symmetrical phase angles with respect to a resonancepoint on the phase angle characteristic curve obtained through thevibration in the liquid being tested.